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   <head>
      <title>Subgroup Info</title>

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      <script src="./SubgroupInfo.js"></script>
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   <body>
      <template id="header_template">
         <h3>${MathML.sans('<mtext>Results of &#8220;Subgroup&#8221;computation for&nbsp;</mtext>' + group.name)}</h3>
         <p>All <a href="./help/rf-groupterms/index.html#subgroup">subgroups</a> of
            ${MathML.sans(group.name)} are listed in the following table. Also, a link
            which exhibits the embedding for each subgroup is also provided, or in
            some cases a reason why it is not possible right now. Subgroup
            <a href="./help/rf-groupterms/index.html#order-of-a-subgroup">order</a> and
            <a href="./help/rf-groupterms/index.html#index-of-a-subgroup">index</a> are also given.</p>
         <p class='gapcode' data-built-in-code-type='all subgroups'></p>
         <p>The column entitled "normal" reports whether the subgroup is
            <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>
            in ${MathML.sans(group.name)}, and if so, tries to provide a link to
            showing that normality by means of a sheet showing a
            short exact sequence (SES) which exhibits the
            <a href="./help/rf-groupterms/index.html#first-isomorphism-theorem">First Isomorphism Theorem</a>
            applied to the subgroup.</p>
         <p class='gapcode' data-built-in-code-type='is normal'></p>
         <p>The subgroups can also be shown arranged in a
            <a href="./help/rf-groupterms/index.html#lattice-of-subgroups">lattice</a>, each shown as
            highlighted portions of the whole group, connected by the identity (inclusion)
            homomorphism. You may see that lattice by
            <a href="javascript:showSubgroupLattice('CDElement')">Cayley diagram</a>,
            <a href="javascript:showSubgroupLattice('CGElement')">cycle graph</a>, or
            <a href="javascript:showSubgroupLattice('MTElement')">multiplication table</a>, or</p>
         <p class='gapcode' data-built-in-code-type='subgroup lattice'></p>
         <p id="simple">None of the subgroups on the list below is
            <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>.
            For this reason, ${MathML.sans(group.name)} is a
            <a href="./help/rf-groupterms/index.html#simple-group">simple</a> group.</p>
         <p id="not-simple">At least one of the subgroups on the list below is
            <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>.
            For this reason, ${MathML.sans(group.name)} is not a
            <a href="./help/rf-groupterms/index.html#simple-group">simple</a> group.</p>
         <p class='gapcode' data-built-in-code-type='is simple'></p>
         <p></p>
         <table id="subgroup_table">
            <thead>
               <tr>
                  <th>Image</th>
                  <th>Name</th>
                  <th>Order</th>
                  <th>Index</th>
                  <th>Normal</th>
               </tr>
            </thead>
            <tbody>
            </tbody>
         </table>
         <p id="no_isomorphism_reason">* No group
            <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a>
            to this subgroup appears in the group library. This makes it
            impossible to draw embedding diagrams, or examples of the
            <a href="./help/rf-groupterms/index.html#first-isomorphism-theorem')">First Isomorphism Theorem</a>.
            You can choose what groups to include in your group library in the
            <a href="">options window</a>.</p>
         <p id="no_quotient_group_reason">** No group
            <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a>
            to the quotient of ${MathML.sans(group.name)} by the normal subgroup appears
            in the group library. This makes it impossible to illustrate the quotient
            group. You can choose what groups to include in your group library in the
            <a href="">options window</a>.</p>
      </template>
      <template id="data_row_template">
         <tr>
            <td class="image">image</td>
            <td>${MathML.sans(MathML.sub('H', index))}</td>
            <td>${subgroup.order}</td>
            <td>${subgroup.index}</td>
            <td>${group.isNormal(subgroup) ? 'yes' : 'no'}</td>
         </tr>
         <tr>
            <td colspan="5">
               <p class="intro">${MathML.sans(MathML.sub('H', index))}${optionalDescription} is
                  {&nbsp;${MathML.csList(element_representations)}&nbsp;}.</p>
               <ul class="points">
               </ul>
            </td>
         </tr>
      </template>
      <template id="isomorphism_template">
         <li>It is
            <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a> to
            ${MathML.sans(isomorphicGroup.name)}, and you can see the embedding by
            <a href="javascript:showEmbeddingSheet(${index}, 'CDElement')">Cayley diagram</a>,
            <a href="javascript:showEmbeddingSheet(${index}, 'CGElement')">cycle graph</a>,
            <a href="javascript:showEmbeddingSheet(${index}, 'MTElement')">multiplication table</a>,
         </li>
      </template>
      <template id="no_isomorphism_template">
         <li class="no_isomorphism">I cannot show you the embedding
            right now (see <a href="#no_isomorphism_reason">below*</a>).
         </li>
      </template>
      <template id="quotient_group_template">
         <li >See the <a href="./help/rf-groupterms/index.html#short-exact-sequence">short exact sequence</a>
            exhibiting the
            <a href="./help/rf-groupterms/index.html#quotient-group">quotient group</a>,
            isomorphic to ${MathML.sans(isomorphicGroup.name)}, by
            <a href="javascript:showQuotientSheet(${index}, 'CDElement')">Cayley diagram</a>,
            <a href="javascript:showQuotientSheet(${index}, 'CGElement')">cycle graph</a>,
            <a href="javascript:showQuotientSheet(${index}, 'MTElement')">multiplication table</a>,
         </li>
      </template>
      <template id="no_quotient_group_template">
         <li class="no_quotient_group">I cannot show you the quotient group
            right now (see <a href="#no_quotient_group_reason">below**</a>).
         </li>
      </template>
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